1. Field of the Invention
The present invention relates to a method and a program for producing data of an original plate (hereinafter referred to as an “original”).
2. Description of the Related Art
Recently, higher resolution has been demanded in a projection exposure apparatus for exposing a circuit pattern drawn on an original (i.e., a mask or a reticle) to a wafer through a projection optical system. Known methods for achieving high resolution includes one using a projection optical system with a high NA, another one using a shorter wavelength for exposure, and still another one for reducing the so-called k1 factor. The following description is focused on the method for reducing the k1 factor.
A circuit pattern is inter alia divided into a line (wiring) pattern and a contact hole pattern. In general, exposing a small contact hole pattern is more difficult than exposing a fine line pattern.
For that reason, various exposure methods have been tried to realize exposure of the contact hole pattern. One typical method is to insert an auxiliary pattern having a sub-resolution size in a mask on which the contact hole pattern to be transferred is drawn.
For example, Robert Socha, Douglas Van Den Broeke, Stephen Hsu, J. Fung Chen, Tom Laidig, Noel Corcoran, Uwe Hollerbach, Kurt E. Wampler, Xuelong Shi, and Will Conley, “Contact Hole Reticle Optimization by Using Interference Mapping Lithography (IML™)”, Proceedings of SPIE, USA, SPIE press, 2004, Vol. 5377, pp. 222-240 (Non-Patent Document 1) and Japanese Patent Laid-Open No. 2004-221594 (Patent Document 1) disclose a technique for deriving, by numerical calculations, in what position an auxiliary pattern is to be inserted. With the disclosed technique, an interference map is obtained by numerical calculations so as to derive a point where interference is caused on the mask and a point where interference is canceled on the mask.
In the point on the interference map where interference is caused, an auxiliary pattern is inserted so that the phase of exposure light having passed through openings of the contact hole pattern to be transferred and the phase of an exposure light having passed through the auxiliary pattern are equal to each other. At the point on the interference map where interference is canceled, an auxiliary pattern is inserted so that the phase of the exposure light having passed through the openings of the contact hole pattern and the phase of the exposure light having passed through the auxiliary pattern have a difference of 180 degrees. As a result, the contact hole pattern to be transferred and the auxiliary pattern strongly interfere with each other, whereby the target contact hole pattern can be exposed successfully.
The above-described interference map represents light amplitude on an image plane that is positioned in an imaging relation to a mask plane. According to the imaging theory in a semiconductor exposure apparatus, i.e., the partial coherent imaging theory, however, it is generally known that an aerial image (light intensity) on the image plane (wafer plane) can be calculated, but the light amplitude on the image plane cannot be obtained. For that reason, when the above-described interference map is obtained, the light amplitude on the image plane is derived by approximation as follows.
First, it is assumed that an aerial image is decomposed into N kinds of eigenfunctions (eigenvectors). This approach is called the Sum of Coherent System Decomposition (SOCS Decomposition) or Karhunen-Loeve transform. To execute the SOCS Decomposition, a Transmission Cross Coefficient (TCC), described in detail later, is derived.
The N kinds of eigenfunctions decomposed according to the SOCS Decomposition have positive and negative values. An aerial image can be obtained by summing up the N kinds of eigenfunctions in terms of intensities. Speaking more exactly, as shown in FIG. 2, an aerial image can be obtained by summing up N kinds of functions which are the products of the eigenvalues corresponding to the i-th eigenfunction (i=1-N) and the square of an absolute value of the i-th eigenfunction. In FIG. 2, Φi represents the eigenfunction and λi represents the eigenvalue. Assuming that the coordinate system on the image plane is (x, y), an aerial image I (x, y) can be expressed by the following formula 1:
      I    ⁡          (              x        ,        y            )        =            ∑              i        =        1            N        ⁢                  ⁢                  λ        i            ⁢                                                            Φ              i                        ⁡                          (                              x                ,                y                            )                                                2            
Assuming that a maximum eigenvalue is a first eigenvalue λ1 and an eigenfunction corresponding to the maximum eigenvalue is a first eigenfunction Φ1(x, y), the first eigenfunction Φ1(x, y) has a maximum contribution to formation of the aerial image. Therefore, the formula 1 can be approximated by the following formula 2:I(x,y)≈λ1|(x,y)|2 From the aerial image I(x, y), an approximate amplitude e(x, y) on the image plane can be obtained from the following formula 3:√{square root over (I(x,y))}=e(x,y)≈√{square root over (λ1)}Φ1(x,y)Thus, the light amplitude on the image plane, i.e., the interference map, can be derived by approximation.
Once the interference map for the image plane is derived, an interference map over the entire mask plane can be derived by executing convolution on an assumption that the contact hole pattern contained in pattern data represents a point having an infinitively small size (i.e., a δ function).
The above-described method has a problem in point of accuracy because the aerial image is approximated only by the eigenfunction Φ1(x, y). To improve the accuracy, however, the following formula 4 should not be simply handled as expressing the light amplitude:
      ∑          i      =      1        N    ⁢          ⁢                    λ        i              ⁢                  Φ        i            ⁡              (                  x          ,          y                )            The reason is that, as seen from the following formula 5, the light amplitude on the image plane cannot be reproduced just by summing up the eigenfunctions as they are:
            I      ⁡              (                  x          ,          y                )              ≠            ∑              i        =        1            N        ⁢                  ⁢                            λ          i                    ⁢                        Φ          i                ⁡                  (                      x            ,            y                    )                    Another problem of the above-described method is that a long calculation time is required to obtain the eigenvalue and the eigenfunction. It is therefore not realistic to derive an optimum solution on a trial and error basis while changing parameters, such as a wavelength, NA, and other information regarding an effective light source.
Still another problem resides in the capacity of the computer memory. In calculating the aerial image, the pupil of a projection optical system has to be divided into meshes. As the number of the meshes increases, the calculation accuracy is improved correspondingly. With an increase in the number of the pupil meshes, however, the number of components of the TCC in the form of a four-dimensional matrix is enormously increased and the capacity of a computer memory has to be scaled up.
Thus, the above-described method for deriving the interference map has problems relating to the calculation accuracy, the calculation time, and the computer memory size.